I have UKPN coming around for the second time this Thursday so will report back. I took few readings at a single socket at various times (that's the extent of my electrical ability) this weekend. I got the following:
11am everything off 224v
11.05am combi turned on 211v
17:00 with lights and combi on 214v
Looking at the voltages at 224v you are only 6v under the nominal supply voltage of 230v turning the combi on drops the voltage by 13v which is quite significant but you don't give any details of the boiler or any other load. When checking the voltages it is a good idea to take load and no load readings as the 17:00 figure is a bit meaningless without having a similarly timed no load reference
I looked into electric boilers for a job a while ago and some of them would push any normal household supply well beyond it's capacity with loads of 15+KW.
Before making any comment about what UKPN are recommending it would need the OP to furnish a lot more info, I would suspect that the property and surrounding properties are on an old overhead supply that predates today's high electricity usage if the supply is feeding a number of properties similarly equipped with electric boilers then it is likely it will not capable of delivering what is needed without reinforcement
The problem the OP has is not unusual these days any will become a major issue if electric vehicles become the norm, I wonder how many installations exceed their original installed capacity.
Only last week I found a 60A cutout that was a bit worse for wear and had clearly been overloaded this was replaced by the DNO with a nice new 100A cut out, yes it gives the installation a bit more capacity but the supply cable is still the same old PILC it was before
My thoughts with the DNO suggestion of an upgrade to three phase would be how close is the three phase to the property and is the DNO trying to get some unsuspecting customer to pay for a network upgrade that ultimately benefits all the customers on that leg of the network.
The OP may benefit from having three phase installed as most electric boilers have multiple elements and the load could be spread across the phases and reduce the limitations that high load single phase appliances can have on an installation